(b) Find the right-hand and the le -hand derivatives of
f
at
x
=
.
(c) Is
f
di erentiable at
x
=
?
. Let
g
be de ned by
g
(
x
)
=
x
-
if
x
≤
x
+
if
x
>
(a) Is
g
continuous at
x
=
?
(b) Is
g
di erentiable at
x
=
?
. Let
f
(
x
)
=
x
+
.
(a) Is
f
continuous at
x
=
?
(b) Is
f
di erentiable at
x
=
?
In
Exercises
-
use the limit de nition of the derivative
f
′
(
c
)
=
lim
h
→
f
(
c
+
h
)
-
f
(
c
)
h
to
evaluate the requested derivative.
. Find
f
′
( )
where
f
(
x
)
=
x
-
x
+
.
. Find
dy
dx
x
=
where
y
=
√
x
+
.
. Find
g
′
( )
where
g
(
x
)
=
x
+
x
-
.
.
Using the alternate de nition of the derivative
f
′
(
c
)
=
lim
x
→
c
f
(
x
)
-
f
(
c
)
x
-
c
,
nd
f
′
(
c
)
where
f
(
x
)
=
x
.
. Find
f
′
(
x
)
where
f
(
x
)
=
x
-
x
+
.
.
Show that
(
,
)
is on the graph of
f
(
x
)
=
√
x
+
and
nd the equation of the tangent line to
the graph at this point.
.
Show that
(
,
-
)
is on the graph of
g
(
x
)
=
x
+
x
-
and
nd the equation of the tangent line
to the graph at this point.
Calculus I
©
-
J. E. Franke, J. R. Griggs and L. K. Norris
Last update: June
,
CHAPTER
. THE DERIVATIVE
. . DEFINITION OF THE DERIVATIVE OF A FUNCTION
.
e volume of a spherical hot air balloon
V
(
r
)
=
π
r
changes as its radius changes.
e
radius is a function of time given by
r
(
t
)
=
t
. Find the average rate of change of the volume
with respect to
t
as
t
changes from
t
=
to
t
=
.
en
nd the instantaneous rate of change
of the volume with respect to
t
at
t
=
.
.
e position function (in feet) of an object dropped from the top of a building
tall as
a function of time (in seconds) is
s
(
t
)
=
-
t
. Find the velocity of this falling object
at
t
=
sec. How long does it take the object to hit the ground, and what is its velocity as it
impacts the ground?
Calculus I
©
-
J. E. Franke, J. R. Griggs and L. K. Norris
Last update: June
,
CHAPTER
. THE DERIVATIVE
. . DEFINITION OF THE DERIVATIVE OF A FUNCTION
. .
Answers to Selected Exercises
.
a)
Yes
b)
f
′
(
+
)
=
-
,
f
′
(
-
)
=
c)
No
.
a)
Yes
b)
No
.
.
c
.
y
-
=
(
x
-
)
.
Average rate of change is
π
.
dV
dt
( )
=
π
Calculus I
©
-
J. E. Franke, J. R. Griggs and L. K. Norris
Last update: June
,
CHAPTER
. THE DERIVATIVE
. . BASIC DIFFERENTIATION RULES
.
Basic Di erentiation Rules
Since the de nition of the derivative involves a limit, we can use our results about limits of functions
from Chapter
to construct some basic results for derivatives. Some of the results about limits
translate directly to statements about derivatives, while others lead to modi ed results.
Example
.
Use the limit de nition of the derivative to compute the derivative of the linear function
f
(
x
)
=
ax
+
b
at an arbitrary point x
=
c and
nd f
′
(
x
)
.
Solution:
f
′
(
c
)
=
lim
h
→
f
(
c
+
h
)
-
f
(
c
)
h
=
lim
h
→
(
a
(
c
+
h
)
+
b
)
-
(
ac
+
b
)
h
=
lim
h
→
ah
h
=
lim
h
→
a
=
a
Hence the derivative of
f
(
x
)
=
ax
+
b
is the constant function
f
′
(
x
)
=
a
. We recognize that
a
is the
slope of the line that is the graph of
f
.
. .
e Derivative of a Sum of Functions
We learned in Chapter
that the limit of a sum of functions at a point is equal to the sum of the
limits, provided that the individual limits exist. We use this result to prove the following theorem on
the derivative of the sum of functions.
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